A participle is that form of the verb which partakes of the nature both of a Verb and of an Adjective.
A participle phrase will begin with a present or past participle. If the participle is present, it will dependably end in ing. Likewise a regular past participle will end in a consistent ed.
Since, all phrases require two or more words, a participle phrase will often include objects and/or modifiers that complete the thought.
Participle phrases always function as adjectives,adding description to the sentence.
In this case,
Option A is correct. The participle phrase 'Trotting up to the fence' modifies the noun 'horse'. The present participle 'trotting' introduces the participle phrase.
Option B is incorrect as 'that you have an apple or carrot' is a clause having its own subject and predicate.
Options C and D are incorrect as they are devoid of any participle.